Unit name | Calculus of Variations |
---|---|

Unit code | MATH30005 |

Credit points | 10 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2D (weeks 19 - 24) |

Unit director | Dr. Tourigny |

Open unit status | Not open |

Pre-requisites |
MATH20901 Multivariable Calculus and MATH20101 Ordinary Differential Equations 2 |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**Unit Aims**

To introduce students to the calculus of variation, and to illustrate its use in the solution of some elementary problems arising in mathematics and in physics.

**Unit Description**

Some simple problems typical of the subject are as follows: Amongst all the curves joining two given points on a manifold (such as the plane, or the sphere), find the one of shortest length; for a given perimeter, find the planar shape of largest area. Calculus of Variations develops the tools necessary to answer such questions; it is an important branch of optimisation in which the quantity (the functional) to be minimised depends on infinite-dimensional vectors that may for instance represent curves or surfaces. The subject has deep connections with various fields in the natural sciences, including differential geometry, ordinary and partial differential equations, materials science, mathematical biology, etc. It is one of the oldest and yet one of the most used tools for the investigation of problems involving the concept of "free energy". The aims of this course are (1) to cover the basics of the calculus of variations, including the one-variable case, and (2) to illustrate the theory by considering various applications arising in the natural sciences.

**Relation to Other Units**

The unit builds on Calculus 1, Linear Algebra and Geometry 1, and Multivariable Calculus. In terms of the mathematics involved, it is also closely connected with other units that pertain to differential and partial differential equations.

After taking this unit, students will:

- know the basic techniques and results of the calculus of variations
- be able to apply these techniques to solve some problems arising in other areas of science that can be formulated in terms of the minimisation of some functional.

The unit will be taught through a combination of

- synchronous online and, if subsequently possible, face-to-face lectures
- asynchronous online materials, including narrated presentations and worked examples
- guided asynchronous independent activities such as problem sheets and/or other exercises
- synchronous weekly group problem/example classes, workshops and/or tutorials
- synchronous weekly group tutorials
- synchronous weekly office hours

100% Timed, open-book examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

**Recommended**

- I M Gelfand and S.V. Fomin,
*Calculus of Variations*, Prentice-Hall, 2000 - Bruce van Brunt,
*The Calculus of Variations*, Dover, 2010